Statistical mechanics canonical

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

If one denotes the averages over a canonical distribution by (.. . ), then the relation A = U-T S and U = (W) leads to the statistical mechanical coimection to the themiodynamic free energy A ... 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

Statistical mechanics may be used to derive practical microscopic fomuilae for themiodynamic quantities. A well-known example is tire virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition fiinction... 

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

In quantum statistical mechanics where a density operator replaces the classical phase density the statistics of the grand canonical ensemble becomes feasible. The problem with the classical formulation is not entirely unexpected in view of the fact that even the classical canonical ensemble that predicts equipartitioning of molecular energies, is not supported by observation. 

Monte Carlo Methods. Although several statistical mechanical ensembles may be studied using MC methods (2,12,14), the canonical ensemble has been the most frequently used ensemble for studies of interfacial systems. In the canonical ensemble, the number of molecules (N), cell volume (V) and temperature (T) are fixed. Hence, the canonical ensemble is denoted by the symbols NVT. The choice of ensemble determines which thermodynamic properties can be computed. 

In MC methods the ultimate objective is to evaluate macroscopic properties from information about molecular positions generated over phase space. To evaluate average macroscopic properties, p, in the canonical ensemble from statistical mechanics, the following expression is used ... 

Statistical mechanics enables one to express the chemical potential i, for an ideal gas phase system in terms of the spectroscopic properties of individual gas phase molecules. The reader is referred to standard statistical mechanics texts (e.g. D. A. McQuarrie Statistical Mechanics , reading list) for the development of the relationship between the system Helmholtz free energy, A , and the corresponding canonical partition function Qi... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The formation of the transition state from the excited molecule is referred to as a microcanonical process, while the formation of the transition state in conventional TST in Chapter4 and in VTST in Chapter 6 is referred to as canonical process. The terms microcanonical and canonical in statistical mechanics refer respectively to processes at constant energy and processes at constant temperature. 

They would become the stars of Prigoginian statistical mechanics. Their importance lies in the fact that, whenever it is possible to determine these variables by a canonical transformation of the initial phase space variables, one obtains a description with the following properties. The action variables / ( = 1,2,..., N, where N is the number of degrees of freedom of the system) are invariants of motion, whereas the angles a increase linearly in time, with frequencies generally action-dependent. The integration of the equations... 

The JE (Eq. (40)) indicates a way to recover free energy differences by measuring the work along all possible paths that start from an equihbrium state. Its mathematical form reminds one of the partition function in the canonical ensemble used to compute free energies in statistical mechanics. The formulas for the two cases are... 

We have been teaching physical chemistry with a kinetics-first orientation for 13 years. Over the course of this past decade we have also examined our students using the ACS Comprehensive Standardized Exam. Form 1995 (16) was used from 1996 through 2003 and Form 2002 (17) was used in 2004 and 2005. This comprehensive exam has been administered at the end of physical chemistry II covering quantum chemistry and spectroscopy. Both versions divide the 60 multiple choice questions into the three canonical areas, T, Q, and D, with the 1995 version assigning 20 questions each to the three areas. The 2002 version assigns 15 questions to the dynamics section and 25 to the quantum section. A few statistical mechanics questions are also scattered among these sections. 

In order to deal with collections of molecules in statistical mechanics, one typically requires that certain macroscopic conditions be held constant by external influence. The enumeration of these conditions defines an ensemble . We will confine ourselves in this chapter to the so-called canonical ensemble , where the constants are the total number of particles N (molecules, and, for our purposes, identical molecules), the volume V, and the temperature... 

Just as there is a fundamental function that characterizes the microscopic system in quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental function having equivalent status, and this is called the partition function. For the canonical ensemble, it is written as... 

Obviously Eq. (11) can also be derived directly from equilibrium thermodynamic considerations involving equilibrium constants. A statistical-mechanical derivation4 of Eq. (11), utilizing an equilibrium (actually grand canonical) ensemble, points up the analogy between this transition and a special case of helix-coil transition, the latter being most usually treated with equilibrium ensembles. 

We take a rather simple approach to the treatment of statistical thermodynamics and the derivations that follow. A rigorous discussion requires introduction of the canonical ensemble and is beyond the scope of our immediate interest. The details can be found, for example, in textbooks on statistical mechanics [269]. The approach taken here yields all of the needed results, and is a compact introduction to the theory. 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

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